On the adjacent vertex-distinguishing total chromatic numbers of the graphs with Δ(G)=3

Let G = (V(G), E(G)) be a simple graph and T (G) be the set of vertices and edges of G. Let C be a k-color set. A (proper) total k-coloring f of G is a function f : T (G) -> C such that no adjacent or incident elements of T (G) receive the same color. For any u is an element of V(G), denote C(u) = {f (u)} boolean OR {f (uv)vertical bar uv is an element of E(G)}. The total k-coloring f of G is called the adjacent vertex-distinguishing if C(u) not equal C(v) for any edge uv. E(G). And the smallest number of colors is called the adjacent vertex-distinguishing total chromatic number chi(at) (G) of G. In this paper, we prove that chi(at) (G) = 6 for all connected graphs with maximum degree three. This is a step towards a conjecture on the adjacent vertex-distinguishing total coloring.